Transcribed TextTranscribed Text

10. Prove the following generalization of Proposition 2.20. Let G and 2 be open in C and suppose f and h are functions defined on G, g : 2 C and suppose that f (G) C S. Suppose that g and h are analytic, 8' (w) + 0 for any w, that f is continuous, h is one-one, and that they satisfy h(z) = 8(f(2)) for Z in G. Show that f is analytic. Give a formula for f'(z). 11. Suppose that f: G C is a branch of the logarithm and that n is an integer. Prove that z" = exp (nf(z)) for all Z in G. 12. Show that the real part of the function 2/2 is always positive. 13. Let G = C - { Z € R : Z V 0} and let n be a positive integer. Find all analytic functions f: G C such that Z = (f(z))" for all Z € G. 14. Suppose f: G C is analytic and that G is connected. Show that if f (z) is real for all Z in G then f is constant. { 1 15. For r > 0 let A = w : w = exp - where 0 < 1/11 < r ; determine the Z set A. G maximal? Are f and g analytic? 17. Give the principal branch of V 1 - z. 19. Let G be a region and define G* = If f: G C is analytic prove that f * : G* C, defined by f* *(z) = f(z), is also analytic.

Solution PreviewSolution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

    By purchasing this solution you'll be able to access the following files:

    for this solution

    or FREE if you
    register a new account!

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Complex Analysis Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats